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Feb 5, 2019 at 11:02 comment added Stefan Waldmann thanks. But be careful, the above check of smoothness is not completely harmless and you should really go through it first :)
Feb 5, 2019 at 10:34 comment added Alex M. This is really interesting, let me see where it takes me (if anywhere). My first thought would be: why not modify the above construction, and instead of the whole convolution thing work with the Fourier transform on $\mathbb R$, in the hope of being able to use the Paley-Wiener theorem? Let me see if I can squeeze anything out of your idea.
Feb 5, 2019 at 10:33 history edited Stefan Waldmann CC BY-SA 4.0
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Feb 5, 2019 at 7:32 history answered Stefan Waldmann CC BY-SA 4.0